Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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1answer
357 views

Radius of Convergence and the Frobenius Method

Consider the equation $$4xy'' + 2y'+ y = 0$$ I know that $x=4$ is a regular singular point, and in the notation that my uni uses, we say that: $$(x-x_0)^2 y'' + (x-x_0)p(x)y' + q(x)y = 0$$ where ...
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1answer
40 views

Power series converging at the convergence radius

Let $f(z)=\sum a_nz^n$ be a power series of radius $R$. By Abel' radial theorem, if $f(R)$ converges then $f$ is continuous over real numbers at $R^-$. I had some questions on how that can be ...
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2answers
20 views

how to prove this statement related to radius of convergence

Suppose that the power series $$\sum b_nx^n$$ converges for $|x|$ less than or equal to $1$. Suppose that for some $s$ greater than $0$, $p(x)=0$ for all $|x|$ less than $s$. How to show that ...
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1answer
22 views

What is the radius of convergence of following series?

suppose that $\sum b_n$ is conditionally convergent but not absolutely convergent. What is the radius of convergence of the following power series $p(x)=\sum b_nx^n$?
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1answer
20 views

For what $x,y$ does $\sum_{k,l\ge 0} \frac{(k+l)!}{k!l!} \left| x^ky^l\right|$ converge?

For what $x,y$ does $\sum_{k,l\ge 0} \frac{(k+l)!}{k!l!} \left| x^ky^l\right|$ converge? I think that $\sum_{k,l\ge 0}\left| x^ky^l\right|$ will converge for $|x|<1$ and $|y|<1$ since ...
1
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1answer
21 views

Solving Laguerre coefficients with Integral?

I'm having some difficulty understanding the solution to a particular Laguerre expansion. The problem reads "Expand the term $ e^{-x}$ as a Laguerre expansion, noting the orthogonality of $$ < ...
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1answer
24 views

how to write as geometric series $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ [on hold]

How would I write $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ as a sum of geometric series?
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1answer
60 views

Find the radius of convergence and interval of convergence of the series

Find the radius of convergence and interval of convergence of the series: $\sum_{n=1}^{\infty}n^n x^{n^4}$ I'm really lost as to how to approach this problem. The other power-series problems were ...
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0answers
43 views

In what sense 1 + 0 + 2 + 0 + 3 + 0 + … = 1/24?

We know that the series $$ 1+2+3+\cdots=-\frac{1}{12} ~~ (1) $$ and $$ 0+1+0+2+0+3+\cdots=-\frac{1}{12} ~~ (2) $$ belong to the elementary Ramanujan class $R=4$ (definition, also here) and the series ...
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1answer
24 views

Function whose $n$-th derivative at $x=0$ is $n^3$ / evaluating the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$

I have troubles proving that the power series $\sum_{n=1}^\infty n^3\frac{x^n}{n!}$ represents the function $f(x)=e^x(x^3+3x^2+x)$. My idea was using the identity theorem for power series and the ...
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0answers
50 views

Prove that the field of Puiseux series over $\mathbb C$ is algebraically closed

Denote by $K=\mathbb{C}((z))$ the fraction field of $\mathbb{C}[[z]]$. Define an embedding of $K$ onto itself taking $a(z)$ to $a(z^n)$ $\forall n$. The target is $\mathbb{C}((z^{1/n}))$. Define the ...
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3answers
1k views

Puiseux series over an algebraically closed field

Using the construction $R_n = K[t^\frac1n]$, $L_n = \text{Quot}(R_n)$ and $P = \bigcup_{n\in \mathbb{N}}L_N$ one automatically gets that the Puiseux series are a field. Nevertheless they are also an ...
1
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1answer
17 views

How does the speed of convergence of these formulae for calculating PI compare with the best algorithms?

I came across some series many years ago for calculating PI. I found that the first member of that series has been known for a long time in the math world. It is the set of series defined by: $$ ...
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4answers
56 views

Show the given series is a solution of $y''-xy'-y=0$

My problem is this: "Show that the function represented by the power series, $$y=\sum_{n=0}^{\infty} \frac{x^{2n}}{2^nn!}=1+ \frac{x^2}{2}+ \frac{x^4}{8}+ \frac{x^6}{48}+...$$ is a solution of the ...
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1answer
106 views

The completness of ring and its power series ring

Let $R$ be a ring and $I$ an ideal of $R$. If $R$ is $I$-adically complete, why then $R[[x]]$ is $(IR[[x]]+(x)R[[x]])$-adically complete? (Matsumura, Commutative Ring Theory, Exercise 8.6.) Take ...
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0answers
71 views

A closed form for the following Series

I was computing some calculations, when I got stuck about a possible closed form for this series: $$S = \sum_{k = 2}^{N}\ \frac{k!}{k^k - k!}$$ I proved by hands that it's absolutely convergent by ...
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1answer
27 views

Show that entire function $f$ is a polynomial of degree at most $n$

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a entire function. Suppose that there are $M$, $r>0$ and $n\in \mathbb{N}$ such that $\left|f(z)\right|<M\left|z\right|^n$ for all $z \in \mathbb{C}$ ...
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4answers
504 views

Finding the power series for $y$ where $y + \sin(y) = x$

What do you do to find the power series for an inverse relationship such as for $y$ in $y + \sin(y) = x$? I'm not sure where to begin. (Similarly, the Lambert $W$ function has such a power series ...
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0answers
29 views

In a recurrence relation, how do we know which order to terminate?

By employing Frobinious or Power Series approach, we my come up with a recurrence relation that is only solvable if we set any constant lower than $a_0$ or higher than $a_n$ vanish. For example, in ...
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0answers
25 views

Lang's proof of the Weierstrass preparation theorem

Relevant Google Books link. I'm having problems with the final step in the proof of Theorem 9.1. It's not clear to me why the function $I + \tau \circ \frac{\alpha(f)}{\tau(f)}$ should be ...
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1answer
642 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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0answers
43 views

Can all series in the elementary Ramanujan class R = 2 be shifted?

For $f(x)=\sum_{n=0}^\infty a_nx^n$ and $g(x)=f(x)-Rf(x^2)$, $R\neq1$, $f(1)$ belongs to the elementary Ramanujan class $R$ if $g(1)$ is Abel summable. The elementary Ramanujan sum of $f(1)$ is ...
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1answer
457 views

Why $e^{\pi}-\pi \approx 20$, and $e^{2\pi}-24 \approx 2^9$?

This was inspired by this post. Let $q = e^{2\pi\,i\tau}$. Then, $$x := \left(\frac{\eta(\tau)}{\eta(2\tau)}\right)^{24} = \frac{1}{q} - 24 + 276q - 2048q^2 + 11202q^3 - 49152q^4+ \cdots\tag1$$ ...
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0answers
31 views

Show that 1 + 2 + 0 + 4 + 0 + 0 + 0 + 8 + … = -1.

The diluted series of powers of $2$ $1+2+0+4+0+0+0+8+\cdots$ belongs to the elementary Ramanujan class $R=2$ and is summable to $-1$ (definition, also here). How to prove that result given ...
1
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1answer
114 views

Generalization of Maclaurin series?

The Maclaurin series for a function $f$ is $$f(x)=\sum_{n=0}^\infty\frac{f^{(n)}(0)x^n}{n!}$$ Suppose that instead of the $x^n$ we picked up a function $g_n$? We can write ...
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2answers
26 views

If $a_k\ge 0$ for all $k$ show that $\sum\limits_{k=0}^na_k\le b\in\mathbb{R}$.

Given that $\lim\limits_{x\rightarrow1^-} \sum\limits_{k=0}^\infty a_kx^k = b \in\mathbb{R}$ for $|x|<1$. If $a_k\ge 0$ for all $k$ show that $\sum\limits_{k=0}^na_k\le b$. This is just a step in ...
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0answers
22 views

Neat expression for finite series with poisson distribution

I have the following expression $$ \sum_{n=1}^N f(k, n, p)\frac{1}{n} $$ where $f()$ is the binomial probability mass function: $$ f(k, n, p) = {n \choose k} p^k (1-p)^{n-k}$$ I wonder whether ...
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1answer
23 views

Converse of Abel's theorem

I know that a non conditional converse of Abel's theorem is not true, but is there a proof for the converse given certain conditions. So if $f(x)=\sum_{k=0}^\infty a_kx^k$ converges when $|x|<1$ ...
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2answers
16 views

Radius of convergence for complex power series

I am supposed to find the radius of convergence for the complex power series $$\sum_{n=0}^{\infty}(-1)^n2^nz^{2n+2}$$ I know that the radius of convergence is calculated by ...
2
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3answers
1k views

Confused by Laurent series

A typical problem related to Laurent series is this: For the function $\frac 1{(z-1)(z-2)}$, find the Laurent series expansion in the following regions: $\\(a) |z|<1, \\ (b) 1<|z|<2, ...
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0answers
26 views

Is the composition of an harmonic function with an analytic function an harmonic function in any dimension?

I was wondering if it is true or not that, given a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ and $g:\Omega\subset\mathbb{R}^n\rightarrow \mathbb{R}^n$ such that $f$ is harmonic and $g$ real ...
3
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2answers
38 views

True or false.If the series converges for x=1.1, then it converges for x=7

I saw this question in a previous year test and it seemed pretty simple, and that can often mean that I am missing something. If the series $$\sum_{n=0}^{\infty}a_n(x-3)^n$$ converges for $x=-1.1$, ...
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3answers
68 views
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2answers
41 views

Limit of power series with L'Hospital

Calculate the given limit: $$\lim_{x\to 0} \frac{1}{1-\cos(x^2)}\sum_{n=4}^\infty\ n^5x^n$$ First, I used Taylor Expansion (near $x=0$): $$1-\cos(x^2)\approx 0.5x^4$$ I'm now quite stuck with the ...
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2answers
58 views

What is the minimal correction to the harmonic series such that it converges?

as you all hopefully know, the series $$ \sum_{k\ge 1}\frac{1}{k} $$ diverges. Now I know that you can add some logarithmic corrections, such that it converges: $$ \sum_{k\ge 1}\frac{1}{k\log(k)^2} $$ ...
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0answers
41 views

Can you make a power series for $y=x^2$?

I tried to make a power series for $y=x^2$ by starting with $f^{-1}(x)=\sqrt{x}$ and applying Lagrange Inversion theorem with $a=1$, but it didn't converge. In fact, the best you could observe from ...
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0answers
22 views

What's the intuition behind this example of a power series converging everywhere on the boundary but not absolutely?

The example is $$\sum_{i=1}^\infty a_i z^i \text{ where } a_i = \frac{(-1)^{n-1}}{2^nn}\text{ for }n=\lfloor\log_2(i)\rfloor+1\text{, the unique integer with }2^{n-1}\le i < 2^n$$ It seems that ...
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1answer
19 views

How To Determine The Radius of This Power Series

$$ \sum_{n\ge 0} (3+\cos n)x^n ; a_n = (3+\cos n) $$ I used d'Alembert : $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n} = \lim_{n\to\infty} \frac{3+\cos(n+1)}{3+\cos n} $$ Nw I'm stuck With How To get Rid ...
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0answers
22 views

$\sum\limits_{k=0}^{\infty}a_k(z+4)^k$ with $a_{2j}=(\sqrt{3})^{2j}$, different solutions

I want to calculate the radius of convergence of the series $$\sum\limits_{k=0}^{\infty}a_k(z+4)^k$$ where $a_{2j}=(\sqrt{3})^{2j}$ and $a_{2j-1}=\frac{1}{2j-1}$. I would calculate the radius of ...
3
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1answer
21 views

Convergence radius and two-times-differentiability of power series.

I wanted to compute the radius of convergence for the following the power series $$\sum_{n=1}^{\infty} a_nz^n$$ with $(i) \, a_n = n!, \, (ii) \, a_n = \sqrt[\leftroot{-3}\uproot{3}n]{n}$ Then I ...
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3answers
42 views

Obtaining the value of a power series similar to sine

I apologies for the vague title and the very specific question. I would like to know what $$K=4\left[ \frac{1}{1\cdot2!}-\frac{1}{3\cdot4!}+\frac{1}{5\cdot6!}-\cdots \right]$$ evaluates to. This is ...
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2answers
27 views

Simplifying Power Series as a Summation - Alternating Coefficients

I'm currently trying to rewrite a power series I have into summation notation. The series is as follows: $$ 2x + 3x^{4} + 2x^{7} + 3x^{10} + 2x^{13} + ... $$ Obviously I'll have $x^{3n+1}$ in the ...
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1answer
28 views

Radius of convergence of the solutions of the differential equation

Justifies that the solutions are analytic functions in $t_0=0$ . Is it possible to determine the radius of cnvergencia series corresponding powers without calculate? $$ (1-t^2)x''-2tx'+a(a+1)x=0$$ ...
6
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2answers
133 views

Radius of convergence of $\sum\limits_{n \ge 1} a_n z^n$ where $a_n$ is the number of divisors of $n^{50}$

Consider the power series $\sum_{n \ge 1} a_n z^n$, where $a_n$ is the number of divisors of $n^{50}$. What is its radius of convergence? My attempt $a_n < n^{50}$ $\forall$ $n$. So $\lim ...
1
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1answer
23 views

An elementary introduction to Puiseux series?

While studying Analytic combinatorics of Flajolet and Sedgewick (to be more specific, the coefficient asymptotics of algebraic functions), I have come across the concept of Newton-Puiseux expansions. ...
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4answers
48 views

How do I evaluate this series?

How do I evaluate this series: \begin{equation} \sum_{n=2}^\infty \frac{\prod_{k=1}^{n-1} (2k-1) }{2^nn!} = \frac{1}{8} + \frac{1}{16} + \frac{5}{128} + \frac{7}{256} +\ldots \end{equation} I ...
2
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1answer
25 views

Find the Taylor series and evaluate at $f^{39}(0)$

$$e^{-x^2}$$ I've had a hard time understanding power series since as long as I can remember. To my understanding, the question is asking me to write out the terms in the formula for Taylor series, ...
0
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1answer
42 views

What is known of convergence and divergence of the following series?

Let the serie $\sum_{k \geq 0} a_k (z-i)^k$ converge for $z = 4$ and diverge for $z=-8$. What is known of convergence and divergence of the following series? (a) $\sum_{k \geq 0} a_k (1+i)^k$ (b) ...
0
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3answers
55 views

Prove that $\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$ converges [closed]

Prove that the following power series converges: $$\sum_{n=1}^\infty \frac{x^n}{n(n+1)}$$ I have tried using d'Alembert's ratio test however this was inconclusive. Anyone have any ideas?
1
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2answers
84 views

Over an integral arising from Kepler's problem [also: generally useful integral, NOT DUPLICATE!]

This post might appear as a duplicate of the following: Over an integral arising from Kepler's problem [also: generally useful integral] So recalling quickly: $$\Phi(\epsilon) = ...